Quantum Gravity PhenomenologyA systematic approach

Yuri Bonder

Departamento de Gravitacion y Teorıa de CamposInstituto de Ciencias Nucleares

Universidad Nacional Autonoma de Mexico

XII Taller de la DGFM-SMF1 de diciembre de 2017

Guadalajara, Jalisco

Theoretical physics status

• Fundamental physics = GR + QM.

• Accurate empirical description (where we have access).

• Theoretically inconsistent ⇒ new theory (QG).

• Towards QG: top down vs. bottom up.

• No clues on the nature of QG!

(Idealized) phenomenologists’ workflow

Parametrizeall violations

Selecta principle

Phenomenology

Put bounds

Experiments

Paradigmchange

No violation

ViolationSelf-consistent?No Yes

• Often, steps 2 and 3 not considered.

Select a principle

Parametrizeall violations

Selecta principle

Phenomenology

Put bounds

Experiments

Paradigmchange

No violation

ViolationSelf-consistent?No Yes

GR principles

• Equivalence principle(s).

• Diffeomorphism invariance.

• Local Lorentz invariance.

• Einstein-Hilbert action.

• Torsion-free.

• 4 dims.

•...

• These principles are not independent.

• In addition, we have the principles of quantum mechanics andthe SM.

Lorentz invariance

• As an example, we focus on local Lorentz invariance.

• Lorentz invariance = all local inertial frames are equivalent.

• Inertial ↔ free particles (w.r.t. known interactions).

• No preferred (nondynamical) spacetime directions.

• At the level of the action: inv. under local SO(1, 3)“rotations” (tetrads).

• Motivation:• LI is fundamental for both GR and QFT.• LV includes CPT violation1.• Motivated by spacetime discreteness.• Accommodated by most QG candidates (e.g., ST, LQG).• Possible discovery of new interactions.• Clear phenomenology: perform the same experiment in

different frames.

1Greenberg PRL 2002

Parametrize all violations

Parametrizeall violations

Selecta principle

Phenomenology

Put bounds

Experiments

Paradigmchange

No violation

ViolationSelf-consistent?No Yes

Effective field theory

• EFT is useful when the fundamental d.o.f. are unknown.

• Requires knowing the field content and symmetries.

• Field content = standard physics;symmetries = standard physics without LI.

• Result: Most general parametrization!Lagrange density1

L = LGR + LSM + LLV.

where LLV contains all possible LV additions to SM + GR.

• Naive expectation: LLV is suppressed by EEW/EP ∼ 10−17.

• Terms of every dimensionality (higher dimensions moresuppressed).

1“Standard Model Extension”: Colladay+Kostelecky PRD 1997; PRD 1998;Kostelecky PRD 2004;. . .

Example: Free Dirac spinor minimal sector in flatspacetime

• Minimal = operators of renormalizable dimension:

L =i

2ψΓµ∂µψ −

i

2(∂µψ)Γµψ − ψMψ,

Γµ = γµ − ηµνcρνγρ − ηµνdρνγ5γρ − ηµνeν

−iηµν f νγ5 −1

2ηµνgρσνσ

ρσ,

M = m + im5γ5 + aµγµ + bµγ5γ

µ +1

2Hµνσ

µν .

• Γµ and M are the most general matrices (e.g., m5).

• SME coefficients: aµ, bµ, cµν , dµν , eµ, f µ, gµνρ,Hµν .

Phenomenology

Parametrizeall violations

Selecta principle

Phenomenology

Put bounds

Experiments

Paradigmchange

No violation

ViolationSelf-consistent?No Yes

Experiments and bounds

Experiments (partial list)

• Accelerator/collider.

• Astrophysical observations.

• Birefringence/dispersion.

• Clock-comparison.

• CMB polarization.

• Laboratory gravity tests.

• Matter interferometry.

• Neutrino oscillations.

• Particle vs. antiparticle.

• Resonant cavities and lasers.

• Sidereal/annual variations.

• Spin-polarized matter.

No evidence of LV ⇒ bounds:

“Data Tables for Lorentz andCPT Violation”

Kostelecky+Russell RMP (2011),(’17 version:

arXiv:0801.0287v10)

• > 150 experimental results.

• Best bounds:matter ∼ 10−34 GeV,photons ∼ 10−43 GeV

Gravity SME sector

• Gravity is coupled with SME coefficients (not matter).

• “Minimal” subsector:

LLV =√−gkabcdRabcd

=√−g[−uR + sabRab + tabcdWabcd

].

• Decade long puzzle1: “the t-puzzle.”

• Recently2 found that tabcd is indeed physical.

• Produces cosm. anisotropies during inflation (tensor modes).

• CMB data (BB angular power spectrum): t0i0j < 10−43.

• 29 orders of mag. improvement w.r.t. best bounds on sab!

1Kostelecky+Bailey PRD 20062Bonder+Leon PRD 2017

Self-consistent?

Parametrizeall violations

Selecta principle

Phenomenology

Put bounds

Experiments

Paradigmchange

No violation

ViolationSelf-consistent?No Yes

Self-consistency

• Are there theoretical restrictions to rule out LV terms?

• In flat spacetime, few interesting tests.• Field redefinitions: Only some linear combinations of the

coefficient’s components are observable.

• Strong evidence that spacetime is not flat.

• Curved spacetime tests:• Field redefinitions.• Diffeomorphism invariance.• Dirac algorithm and Cauchy problem.• Gravitational d.o.f.• Spacetime boundaries.

Field redefinitions

• ψ → e iaµxµψ shows that aµψγ

µψ is unphysical.

• In flat spacetime, one-to-one correspondence betweencoordinates and vectors.

• This cannot be done in curved spacetime.

• Less field redefinitions ⇒ access more coefficients1.

• No need for curvature, only nonminkowskian coordinates2.

• The metric can be redefined ⇒ alternative constraints3.

1Kostelecky+Tasson PRD 20112Bonder PRD 20133Bonder PRD 2015

Diffeomorphism invariance

• Nondynamical fields break (active) diffeomorphism invariance.

• Thus, ∇aTab 6= 0, which goes against the Bianchi identities!

• Position: LV must be spontaneously broken1.

1Kostelecky PRD 2004

Dirac algorithm and Cauchy problem

• Dirac algorithm: Is there a Hamilton density for which theevolution respects the constraints?

• Cauchy problem:• Is the evolution uniquely determined by proper initial data?• Is the evolution continuous under changes of initial data.• Are the effects of modifying the initial data in agreement with

spacetime causal structure?

• These conditions are difficult to verify without specifying thecoefficients dynamics.

Cauchy problem: concrete model

• Focus on a concrete model1:

L =1

2DµφD

µφ∗ − m2

2φφ∗ − 1

4BµνB

µν − κ

4(BµB

µ − b2)2

• Flat spacetime, complex scalar field φ (matter), real vectorfield Bµ.

• Bµν = ∂µBν − ∂νBµ and Dµφ = ∂µφ− ieBµφ⇒ LLV = −BµJµ and no gauge freedom.

• Generalization of the Mexican hat potential, its VEV istimelike.

• e, κ, and b are real positive constants.

• Canonical momenta:

π0 =δL

δ∂0B0= 0, πi =

δLδ∂0B i

= B i0,

p =δLδ∂0φ

=1

2(∂0φ

∗ + ieB0φ∗) = (p∗)∗.

1Bonder+Escobar PRD 2016

Cauchy problem: concrete model

L =1

2DµφD

µφ∗ − m2

2φφ∗ − 1

4BµνB

µν − κ

4(BµB

µ − b2)2

• Two second-class constraints:

χ1 = π0,

χ2 = ∂iπi − κB0(BµB

µ − b2) + 2eIm(φp).

• The Dirac algorithm exhausted without inconsistencies.

• E.o.m. not of the form where one can use the “initial value”theorems.

• D.o.f.: B i , πi , φ, and p (only this initial data needed)

⇒ the initial B0 obtained through the constraints.

• No unique initial B0 ⇒ ill-posed Cauchy problem!

Cauchy problem: concrete model

• Example (homogeneous): initially B i = 0, πi = 0, φ = 0, andp = a ∈ C.

χ2 =[B0(0)2 − b2

]B0(0) = 0 ⇒ B0(0) = b, 0,−b.

• Numerically (κ = b/MeV2 = e = m/MeV = Re(a)/MeV =Im(a)/MeV = 1):

1 2 3 4 5t/6.6 10-22 s

-2

-1

1

2

ϕ/MeVB0(0)=0

1 2 3 4 5t/6.6 10-22 s

-2

-1

1

2

ϕ/MeVB0(0)=-b

1 2 3 4 5t/6.6 10-22 s

-2

-1

1

2

ϕ/MeVB0(0)=b

where the blue (yellow-dotted) line is for Reφ (Imφ).

• φ represents matter ⇒ physical consequences!

Cauchy problem: concrete model

• Easy fix: change the kinetic term for Bµ... but the Cauchyproblem for gravity can be damaged.

• Alternatives:• “Only one measurement”.... per spatial point (unlike a

fundamental constant).• Consider B0 as a standard d.o.f. (i.e., naive application of

Lagrange’s formalism ⇒ inequivalent quantizations?, discretenumber of d.o.f.).

• Construct a criteria to choose a special B0 (e.g., initial energy,but there are degeneracies).

• Longterm goal: study if we can rule out spontaneous LV.

Gravitational degrees of freedom

• Palatini vs. conventional• For the minimal gravitational LV, the standard and Palatini

approaches are equivalent1.• More general field redefinitions, no practical applications!• For nonminimal LV, these approaches are inequivalent.

• Boundaries• In the phenomenological applications of LV, spacetime is

conformally flat, which has boundaries.• For the minimal gravitational action, add2

∆SLV = ±2

∫boundary

d3x√|h|nµnσkµνρσKνρ.

• Tricky to find ∆SLV for the nonminimal part!

1Bonder PRD 20152Bonder PRD 2015

Conclusions

• Looking for empirical clues of new physics could play animportant role towards QG.

• This must be done systematically: with generality andchecking the self-consistency.

• This type of program has been applied mainly for LV.

• EFT provides the general parametrization.

• Such a parametrization allows us to test LV experimentallyand theoretically.

• New interesting phenomenological connections withcosmological observations.

• Several theoretical restrictions, mainly in curved spacetime.

Paradigm change

Parametrizeall violations

Selecta principle

Phenomenology

Put bounds

Experiments

Paradigmchange

No violation

ViolationSelf-consistent?No Yes

Gibbons-Hawking term

• In the minimal gravitational LV action-variation:

δS ⊃ 1

2κ

∫Md4x√−g(g caδbd + kabcd)δRabc

d

=1

κ

∫Md4x√−g(∇c∇dk

cabd)δgab

+1

κ

∫∂M

d3x√|h|nc(2ga[cgb]d + kcabd)∇dδgab

• In ∂M: δgab = 0 (and δhab = δna = 0) but nc∇cδgab 6= 0.

• Kab = hca∇cnb ⇒ δKab = −hcandδCcbd = 1

2hcan

d∇dδgbc ⇒

nc(2ga[cgb]d+kcabd)∇dδgab = −δ[(

2hab ± 2ncndkcabd

)Kab],

• To cancel the problematic term:

∆S =1

κ

∫∂M

d3x√|h|(

2hbc ± 2nandkabcd

)Kbc .

Variation under diffeomorphisms

• Nongravitational LV: S =∫d4x√−gR + 2κSm(g , φ; k).

• Under a diffeo. assoc. with any ξa (of compact sup.):

δS =

∫d4x

(δ√−gRδgab

δgab + 2κδLmδgab

δgab + 2κδLEHδφ

δφ

)=

∫d4x (−Gab + κTab) (−2∇(aξb))

= 2

∫d4x (−∇aGab + κ∇aTab) ξb

= 2κ

∫d4xξb∇aTab,

where use that the fields φ satisfy their e.o.m.,δgab = Lξgab = −2∇(aξb), and the Bianchi identity.

• Hence, δS = 0 if and only if ∇aTab = 0.

Dirac method

• Dirac’s algorithm: method to construct the Hamiltonian.

pi=dL/dq'i

L=L(qi,q'

i)

ξ' = {ξ,H}=0

H → H + u ξ

Outcomes

Successfultheory

New ξ

Valid by fixing u

Constraints?{ξ} No

Yes

Inconsistency(e.g., 0=1)

Discard thetheory

H=piq'i-L(q

i,pi)

• May reveal inconsistencies (example: L(q, q) = q).

Cauchy theorems

• Cauchy-Kowalewski requires analytic initial data, whichdamages causality.

Theorem

(M, gab) globally hyperbolic, ∇a any derivative operator. Thefollowing system of n linear equations for n unknown functionsΨ1, . . . ,Ψn

gab∇a∇bΨi + Aaij∇aΨj + BijΨj + Ci = 0,

where Aaij , Bij , Ci are smooth vector/scalar fields, has a well-posed

Cauchy problem.

• There are more general theorems1.

• Most relevant: form of the second-derivative term.

1Wald’s GR book